Principal curves are smooth curves that minimize the average squared orthogonal distance to each point in a data set. Fitting a principal curve is a maximum-likelihood technique for nonlinear regression in the presence of Gaussian noise on both x and y. We choose two definitions of principal curves in the literature and then present experimental results to discuss over them.

We use Eisenstein’s irreducibility criterion to prove that there exists an absolutely irreducible polynomial P(X,Y) ∈ GF(q)[X,Y] with coefficients in the finite field GF(q) with q elements, with prescribed level curves Xc := {(x,y)∈GF(q)2 | P(x,y)= c}. 2000 Mathematics Subject Classification. 11T06.

Background: Monte Carlo and experimental relative dose determination in a water phantom, due to a high dose rate (HDR) 192Ir source is presented for real energy spectrum and monochromatic at 356 keV. Materials and Methods: The dose distribution has been calculated around the 192Ir located in the center of 30 cm ×30 cm ×30 cm water phantom using MCNP4C code by Monte Carlo method. Relati...

We solve the equation f(x, y) + g(x, y) = x + y for homogeneous f, g ∈ C(x, y), completing an investigation begun by Viète in 1591. The usual addition law for elliptic curves and composition give rise to two binary operations on the set of solutions. We show that a particular subset of the set of solutions is ring-isomorphic to Z[e].

Using the parametrizations of Kubert, we show how to produce in nite families of elliptic curves which have prescribed nontrivial torsion over Q and rank at least one. These curves can be used to speed up the ECM factorization algorithm of Lenstra. We also brie y discuss curves with complex multiplication in this context.

The purpose of this paper is establish and apply an enumerative formula or 'method' dealing with a family C = { ¯ C y : y ∈ Y } of rational curves on a variety S, e.g. a rational surface. Significantly, the family C is not assumed to be the family of 'all' rational curves of given homology class: rather, we require only that it be sufficiently large (n = dim Y ≥ 3) and well-behaved as regards d...

An Engel curve describes how a consumer’s purchases of a good like food varies as the consumer’s total resources such as income or total expenditures vary. Engel curves may also depend on demographic variables and other consumer characteristics. A good’s Engel curve determines its income elasticity, and hence whether the good is an inferior, normal, or luxury good. Empirical Engel curves are cl...

In a previous paper, we proved that over a finite field k of sufficiently large cardinality, all curves of genus at most 3 over k can be modeled by a bivariate Laurent polynomial that is nondegenerate with respect to its Newton polytope. In this paper, we prove that there are exactly two curves of genus at most 3 over a finite field that are not nondegenerate, one over F2 and one over F3. Both ...

15 Parametric Curves And Surfaces 15.1 Parametric Curves Designing Curves • We don’t want only polygons. • Curves are used for design. Users require a simple set of controls to allow them to edit and design curves easily. • Curves should have infinite resolution, so we can zoom in and still see a smooth curve. • We want to have a compact representation. Parametric functions are of the form x(t)...

Let h:X → Y be a finite morphism of smooth connected complete curves over Cp. We show h extends to a finite morphism between semi-stable models of X and Y . 2000 Mathematics Subject Classification: 11G20